Quadratic Functions and Solutions

Lesson 13

Math

Unit 7

9th Grade

Lesson 13 of 13

Objective


Interpret quadratic solutions in context.

Common Core Standards


Core Standards

  • A.CED.A.1 — Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
  • F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
  • F.IF.C.8.A — Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Criteria for Success


  1. Describe in context what it means to be a solution to a quadratic function.
  2. Interpret points in a situation from a table of values and a graph. 
  3. Describe other features of the quadratic function including vertex, intercepts, and symmetry in context of the situation. 
  4. Identify and describe domain restrictions in context of situations. 
  5. Create quadratic equations in one variable for geometric applications and interpret solutions in geometric context.

Tips for Teachers


  • In regard to pacing, this lesson may be split over two days. It may also be adjusted to take on more of a project-based approach. 
  • Students will encounter more applications of quadratic functions in Unit 8; this lesson serves to bring the concepts and skills of this unit together while also introducing students to some situations they will see in more depth in the next unit.
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Anchor Problems


Problem 1

A ball is thrown straight up from the top of a $${48}$$-foot tall building with an initial speed of $${32}$$ feet per second. The height of the ball as a function of time can be modeled by the function $$h(t)=-16t^2+{32}t+{48}$$. Below is a graph and table of the function.

a. What are the solutions to this function? Interpret each solution in context of the situation. 

b. What is the vertex of this function? Interpret the vertex in context of the situation. 

Guiding Questions

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Problem 2

The table below represents the value of Andrew’s stock portfolio, where $$V$$ represents the value of the portfolio in hundreds of dollars and $$t$$ is the time in months since he started investing. Answer the questions that follow based on the table of values.

$$t$$(months) $$V(t)$$ (hundreds of dollars)
2 325
4 385
6 405
8 385
10 325
12 225
14 85
16 -95
18 -315
  1. Assuming this data is quadratic, how much did Andrew invest in his stock initially? Explain how you arrived at this answer.
  2. What is the maximum value of his stock and how long did it take to reach the maximum value?
  3. After approximately how many months is Andrew’s stock portfolio valued at $0?
  4. How fast is Andrew’s stock value decreasing between months 10 and 12? Find a two-month interval where the average rate of change is faster than it is between months 10 and 12 and explain why.

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 4 > Topic A > Lesson 10Example 2

Algebra I > Module 4 > Topic A > Lesson 10 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 3

John is building a small rectangular pen for chickens. He wants the length to be 6 more feet than twice the width. 

If the area of the pen is 36 square feet, what are the dimensions of the pen in feet?

Guiding Questions

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Problem Set


Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task


A toy company is manufacturing a new toy and trying to decide on a price that maximizes profit. The graph below represents profit, $$P$$, generated by each price of a toy, $$x$$. Answer the questions based on the graph of the quadratic function model.

  1. If the company wants to make a maximum profit, what should the price of a new toy be?
  2. What is the minimum price of a toy that produces profit for the company? Explain your answer.
  3. What does the point $$P(10)=0$$ mean in context of the situation?
  4. Find the domain that only results in a profit for the company and find its corresponding range of profit.

References

EngageNY Mathematics Algebra I > Module 4 > Topic A > Lesson 10Exit Ticket

Algebra I > Module 4 > Topic A > Lesson 10 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include additional application problems around projectile motion; include a graph or table or both in the question for students to interpret
  • Include examples with two projectile motion parabolas graphed in the same coordinate plane and ask students comparison questions about the maximum height, time to reach the ground, etc. 
  • Include area applications similar to Anchor Problem #3 (Do not include problems that ask about maximizing or minimizing; these will appear in Unit 8)
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Lesson 12

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Features of Quadratic Functions

Topic B: Factoring and Solutions of Quadratic Equations

Topic C: Interpreting Solutions of Quadratic Functions in Context

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